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months in which to write out essays on three topics acceptable
to the examiners. The third question inspired a fine dissertation
on the Socratic method in secondary teaching, a method which
Weierstrass followed with brilliant success when he became the
foremost mathematical teacher of advanced students in the
A teacher - at least in higher mathematics - is judged by his
students. If his students are enthusiastic about his 'beautifully
dear lectures', of which they take copious notes, but never do
any original mathematics themselves after getting their
advanced degrees, the teacher is a flat failure as a university
instructor and his proper sphere - if anywhere - is in a secon-
dary school or a small college where the aim is to produce tame
gentlemen but not independent thinkers. Weierstrass' lectures
were models of perfection. But if they had been nothing more
tbftn finished expositions they would have been pedagogically
worthless. To perfection of form Weierstrass added that intan-
gible something which is called inspiration. He did not rant
about the sublimity of mathematics and he never orated; bub
somehow or another he made creative mathematicians out of a
disproportionately large fraction of his students.
The examination which admitted Weierstrass after a year of
probationary teaching to the profession of secondary school
work is one of the most extraordinary of its kind on record. One
of the essays which he submitted must be the most abstruse
production ever accepted hi a teacher's examination. At the
candidate's request Gudermann had set Weierstrass a real
mathematical problem: to find the power series developments
of the elliptic functions. There was more than this, but the part
mentioned was probably the most interesting.
Gudermann's report on the work might have changed the
course of Weierstrass' life had it been listened to, but it made no
practical impression where it might have done good. In a post-
script to the official report Gudermann states that *This pro-
blem, which in general would be far too difficult for a young
analyst, was set at the candidate's express request with the
consent of the commission.* After the acceptance of his written
work and the successful conclusion of his oral examination,